Problem: Evaluate the iterated integral. $ \int_0^1 \left( \int_e^{e^5} \dfrac{x^2}{y} \, dy \right) dx =$ Choose 1 answer: Choose 1 answer: (Choice A) A $1$ (Choice B) B $2$ (Choice C) C $\dfrac{4}{3}$ (Choice D) D $\dfrac{2}{3}$
Solution: Evaluate the inner integral: $\begin{aligned} \int_0^1 \left( \int_e^{e^5} \dfrac{x^2}{y} \, dy \right) dx &= \int_0^1 \left[ x^2 \ln (y) \right]_e^{e^5} dx \\ \\ &= \int_0^1 x^2 (\ln(e^5) - \ln(e)) \, dx \\ \\ &= \int_0^1 4x^2 \, dx \end{aligned}$ Evaluate the outer integral: $\begin{aligned} \int_0^1 4x^2 \, dx &= \dfrac{4x^3}{3} \bigg|_0^1 \\ \\ &= \dfrac{4}{3} \end{aligned}$ The answer: $ \int_0^1 \left( \int_e^{e^5} \dfrac{x^2}{y} \, dy \right) dx = \dfrac{4}{3}$